Domains with a continuous exhaustion in weakly complete surfaces

In previous works, Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and D⊆ X a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X.